Alain Pajor

Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any e > 0, there existsC(e) > 0, suﰁch that if N ∼ C(e)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ e, with probability larger than 1 − exp(−c√n).

The Gelfand widths of l p -balls for 0<p≤1

We provide sharp lower and upper bounds for the Gelfand widths of

Gelfand numbers of operators with values in a Hilbert space

\ell_p

The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces

-balls in the

Volume ratio and other s-numbers of operators related to local properties of Banach spaces

N

The covering numbers and “low *-estimate" for quasi-convex bodies

-dimensional

Euclidean embeddings in spaces of finite volume ratio via random matrices

\ell_q^N

A Short Proof of Paouris' Inequality

-space for

A note on subgaussian estimates for linear functionals on convex bodies

0<p\leq 1

Tail estimates for norms of sums of log-concave random vectors

and